As usual in process algebra, one wishes to coarsen a semantic theory by abstract-ing from internal computation,i.e., the unobservable actionτ which is supposed to be hidden from an external observer. While doing so is usually quite straight-forward for CCS-based calculi [20], it turns out to be highly non-trivial here; this may be the reason why it has not been attempted by Moller and Tofts in [22].
We start off by defining a weak version of our reference preorder, the amor-tized faster-than preorder, which requires us to introduce the following auxiliary notations. For any actionα we define ˆα=df, if α=τ, and ˆα=df α, otherwise.
4Moller and Tofts incorrectly claim in their example that AM+SM∼mtAM [22]. This contra-dicts the correctness of Axiom (P6); AM + SM∼mtAM can be seen directly using Definition 3.1 when matching the only problematic transition AM + SMmail−→ σ10.deliver.0 by the transition sequence AM−→σ 8mail−→σ2.deliver.0and byσ10.deliver.0−→σ 8σ2.deliver.0.
Further, we let =⇒ =df −→τ ∗ and write P =γ⇒ Q, where γ ∈ A ∪ {σ}, if there existR andS such thatP =⇒R−→γ S=⇒Q. We also let=σ⇒0 stand for=⇒.
Definition 7.1(weak amortized faster-than preorder). A family (Ri)i∈Nof rela-tions overP is afamily of weak faster-than relations if, for alli∈N,P, Q ∈ Ri, andα∈ A:
(1) P −→α P implies∃Q, k, k. Q=σ⇒k=αˆ⇒=σ⇒kQ andP, Q ∈ Ri+k+k. (2) Q−→α Q implies∃P, k, k. k+k≤i, P =σ⇒k=αˆ⇒=σ⇒kP and
P, Q ∈ Ri−k−k.
(3) P −→σ P implies∃Q, k≥0. k≥1−i,Q=σ⇒kQ, andP, Q ∈ Ri−1+k. (4) Q−→σ Q implies∃P, k≥0. k≤i+1,P =σ⇒kP, andP, Q ∈ Ri+1−k. We writeP≈iQifP, Q ∈ Ri for a family of weak faster-than relations (Ri)i∈N, and call≈0theweak amortized faster-than preorder.
One can easily check that (≈i)i∈N is the (componentwise) largest family of weak faster-than relations. Moreover, relation≈0 is indeed a preorder; while reflexivity is obvious, establishing transitivity is simple but not trivial. The best way of proving transitivity is by showing thatRk=df{≈i◦≈j|i+j =k}, fork∈N, is a family of weak faster-than relations. This can be done most elegantly by “diagram chasing” as in Figure 1, drawing one diagram per condition of Definition 7.1. In each case, we takeP, Q, R withP≈iQ≈jR(dashed lines) andk=i+j, and we deriveP, R ∈Rk for some suitable k (dotted line).
Our weakening of the amortized faster-than preorder might appear surprising at first sight, due to the presence of =σ⇒k trailing weak action transitions on the right-hand side of the definition. As usual for weak bisimilarity, one may have a number of internal transitions before and after a matching action transition, and to get to these trailing internal transitions one may need to pass further clock transitions.
As in the strong case, it is easy to see that≈0is not a precongruence, even not for parallel composition. To identify the largest precongruence contained in≈0, one may be tempted to first define a straightforward weak variant of the MT-preorder (with Cond. (3’) as on page 600) and hope that this preorder is compositional for all operators except summation. The according definition would impose the following conditions on the notion of a weak MT-relationR ⊆ P × P, forP, Q ∈ R and α∈ A:
(1) P −→α P implies∃Q, k, P, k. Q=σ⇒k=αˆ⇒=σ⇒kQ,P =σ⇒k+kP, and P, Q ∈ R.
(2) Q−→α Q implies∃P. P =α⇒ˆ P andP, Q ∈ R.
(3) P −→σ P implies∃Q, P, k. Q=σ⇒kQ,P=σ⇒k−1P, andP, Q ∈ R. (4) Q−→σ Q implies∃P. P =σ⇒P andP, Q ∈ R.
Unfortunately, this preorder is not even included in ≈0, nor is it included in any other desirable weak faster-than preorder. The reason for this is that, e.g.,
τ* τ* -transition of the allegedly faster process toτ.a.0+τ.b.0can be matched bya.0−→σ a.0and choosingτ.a.0+τ.b.0−→τ a.0−→σ a.0. However,τ.(τ.a.0+τ.b.0)≈0a.0, as the transition sequenceτ.(τ.a.0+τ.b.0)−→τ τ.a.0+τ.b.0−→τ b.0−→b 0cannot be matched by processa.0. This example suggests one to demand, in Condition (1), P −→σ k+kP. Similarly, the exampleσ.(τ.a.0+τ.b.0) andσ.τ.a.0shows that Con-dition (3) should be modified to demand P −→σ k−1P. Furthermore, exploring compositionality for parallel composition implies also in Condition (4)P −→σ P (cf. Proof of Prop. 7.6), which means that we may simply write Q −→σ Q and P, Q ∈ Rin Condition (3) as well. This leads to the following definition of the weak Moller-Tofts preorder.
Definition 7.2(weak MT-preorder). A relationR ⊆ P ×Pis aweak MT-relation if, for allP, Q ∈ Randα∈ A:
We writeP≈mtQ ifP, Q ∈ R for some weak MT-relationR, and call ≈mt the weak MT-preorder.
We first show that≈mt is a preorder. While reflexivity is obvious, it is difficult to see whether ≈mt is transitive,i.e., whether ≈mt◦≈mt ⊆≈mt holds. In order to prove transitivity, we first note that≈mt satisfies a property to which we refer as quasi-transitivity.
Lemma 7.3(quasi-transitivity). ∼mt◦≈mt⊆≈mt.
Proof. We show that ∼mt ◦ ≈mt is a weak MT-relation and restrict ourselves to the most interesting case of establishing Condition (1) of Definition 7.2. Let P, Q, R such that P∼mtQ and Q≈mtR, and let P −→α P for some α ∈ A and P ∈ P. Because ofP∼mtQwe may infer the existence ofQ, Q, k, Psuch that Q−→σ kQ−→α Q,P−→σ kP, andP∼mtQ. Consequently, and by assumption Q≈mtR, there exists process R such that R −→σ kR and Q≈mtR. According to Definition 7.2(1) we may further derive the existence ofR, l, l, Q satisfying R =σ⇒l=α⇒ˆ =σ⇒lR,Q−→σ l+lQ, andQ≈mtR. Definition 3.1(4) then yields P −→σ l+lP for someP withP∼mtQ. Hence we haveR =σ⇒k+l=αˆ⇒=σ⇒l R,P−→σ k+l+lP, and P, R ∈∼mt◦≈mt, as required.
Next we establish an important technical lemma for which we need to introduce some notation. For w, w ∈ (A ∪ {σ})∗ we write w ≡v w if wΛ∪Λ = wΛ∪Λ. Intuitively, w ≡v w if the words w, w are visibly equivalent, i.e., if they are identical up to occurrences ofσ and τ. We also let |w|σ denote the number of occurrences ofσin w.
Lemma 7.4. Let Q, Q, R∈ P andw∈(A ∪ {σ})∗ with Q≈mtR andQ−→w Q. Then there exists someQ, R∈ P,l∈N, andw∈(A∪{σ})∗such thatw≡vw,
|w|σ=|w|σ+l,Q−→σ lQ,R−→w R, andQ≈mtR.
Proof. The proof is by induction on the structure of wordw. If w=, then the statement holds trivially. If w = σv for some v ∈ (A ∪ {σ})∗, then one may easily prove the statement by referring to the induction hypothesis. Hence, we are left with the casew=αv for someα∈ A. Thus, let process ˆQbe such that Q−→α Qˆ −→v Q. By Condition (1) of Definition 7.2, there are processesR,Qˆ, a number ˆl, and a word wα withwα≡v α, |wα|σ = ˆl, R−→wα R, ˆQ−→σ ˆlQˆ, and Qˆ≈mtR. Due to the laziness property in TACSlt, there exists some Q with Q−→σ ˆlQ. We may now apply Lemma 3.3(2) to obtain a process ˆQ satisfying Qˆ −→σ ˆlQˆ−→v QˆandQ∼mtQˆ. Applying the induction hypothesis to ˆQ, v, R yields processes ˆQ, R, a numberl, and a wordvfulfilling the conditionsv≡vv,
|v|σ =|v|σ+l, ˆQ −→σ lQˆ, R −→v R, and ˆQ≈mtR. SinceQ∼mtQˆ and Qˆ−→σ lQˆwe know by Condition (4) of Definition 3.1 of the existence of some
processQ such that Q−→σ lQ andQ∼mtQˆ. Thus,Q∼mtQˆ≈mtR and, by quasi-transitivity,Q≈mtR. By settingw =df wαv andl =df ˆl+l we are
done.
Using this lemma we can now prove the transitivity of the weak MT-preorder.
Proof of property. ≈mt◦≈mt ⊆≈mt. It is sufficient to show that ≈mt◦≈mt is a weak MT-relation. Let P≈mtQ≈mtR for some processes P, Q, R. We focus only on Condition (1) of Definition 7.2, since all other conditions are trivial to establish. LetP −→α P, for which the premise P≈mtQimplies the existence of someQ, k, P, k such that Q =σ⇒k=α⇒ˆ =σ⇒kQ, P −→σ k+kP, and P≈mtQ. Further, we apply Lemma 7.4 to obtain w∈(A ∪ {σ})∗, l∈N,Q∈P, and R∈P such that w ≡v αˆ, |w|σ = k+k+l, Q −→σ lQ, R −→w R, and Q≈mtR. Finally, Condition (4) of Definition 7.2 guarantees the existence of somePsuch that P −→σ lP and P≈mtQ. Hence, R =σ⇒l=αˆ⇒=σ⇒lR for some l, l∈N withl+l=k+k+l, and P≈mtQ≈mtR. It is obvious from Definitions 3.1 and 7.2 that the MT-preorder ∼mt is a weak MT-relation and thus included in the weak MT-preorder≈mt.
Lemma 7.5. ≈mt is included in the weak amortized faster-than preorder≈0. Proof. We prove thatRi=df{P, Q |P −→σ iP≈mtQ}, wherei∈N, is a family of weak faster-than relations. LetP, Q ∈ R, i.e., P −→σ iP andP≈mtQfor some i ∈N and P ∈ P. The only interesting part of the proof concerns establishing Condition 1 of Definition 7.2.
Accordingly, assumeP −→α P for some α∈ A andP ∈ P. Because of the laziness property ofTACSlt, there exists someP1such thatP−→σ iP1. Applying Commutation Lemma 3.3(2) yields a processP2satisfyingP −→α P2andP1∼mtP2. Further, because of P≈mtQ we know of the existence of Q, k, k, P3 such that Q =σ⇒k=αˆ⇒=σ⇒k Q, P2 −→σ k+kP3, and P3
≈mtQ. Moreover, Definition 3.1(4) implies P1 −→σ k+kP4 for some P4 ∈ P with P4∼mtP3. Hence, P −→σ i+k+kP4
andP4∼mtP3
≈mtQ. By quasi-transitivity (cf.Lem. 7.3) and the definition ofR we may now concludeP, Q ∈ Ri+k+k, as desired.
The weak MT-preorder is not only a preorder but also a precongruence.
Proposition 7.6. The weak MT-preorder ≈mt is compositional for all TACSlt operators except for the summation operator.
Proof. We restrict ourselves to the most interesting case of verifying composi-tionality of ≈mt with respect to parallel composition. To do so we show that R=df{P1|P2, Q1|Q2 |P1
≈mtP2, Q1
≈mtQ2}is a weak MT-relation.
LetP1|P2, Q1|Q2 ∈ R be arbitrary. The only difficult part of the proof con-cerns establishing Condition (1) of Definition 7.2 in the case of synchronization.
LetP1|P2−→τ P1|P2for processesP1, P2, due toP1−→a P1andP2−→a P2for some visible action a. Since P1
≈mtQ1 we know of the existence of some Q1, k, P1, k such that Q1 σ
=⇒k a=⇒=σ⇒kQ1, P1 −→σ k+kP1, and P1≈mtQ1. Similarly, since P2
≈mtQ2 we know of the existence of some Q2, l, P2, l such that Q2 σ
=⇒l a=⇒
=σ⇒lQ2,P2 −→σ l+lP2, andP2≈mtQ2. We distinguish the following cases:
• k =l: W.l.o.g. we further assume k ≥l. Due to the laziness property in TACSlt there exists some Q2 with Q2 −→σ k−l Q2 and, because of P2≈mtQ2, there exists some ˆP2such thatP2−→σ k−lPˆ2and ˆP2≈mtQ2. Then,Q1|Q2 σ
=⇒k τ=⇒=σ⇒kQ1|Q2 andP1|P2 −→σ k+kP1|Pˆ2 by our oper-ational rules, andP1|Pˆ2, Q1|Q2 ∈ Rby the definition ofR.
• k = l: W.l.o.g. we assume k > l. We refer to the process between the weak clock transitions and the weak action transition on the path Q2 σ
=⇒l a=⇒=σ⇒lQ2 as ˆQ2. Because of the laziness property in TACSlt and sinceP2≈mtQ2, there exist processes ˆP2,Qˆ2satisfyingP2−→σ k−lPˆ2, Q2−→σ k−lQˆ2, and ˆP2≈mtQˆ2. (This is the place in this proof we referred to in the last few lines before Def. 7.2.) We may now apply Lemma 3.3(2) and Definition 3.1(3) to obtain some ˆQ2 such that ˆQ2−→σ k−l a=⇒=σ⇒lQˆ2 and ˆQ2∼mtQˆ2 . Now, ˆP2≈mtQˆ2∼mtQˆ2, whence ˆP2≈mtQˆ2 because of
∼mt ⊆ ≈mt and the transitivity of≈mt. Now we are in the casek=l. This concludes the compositionality proof of≈mt. As expected for a CCS-based process calculus, ≈mt is not a precongruence for the summation operator, but the summation fix used for other bisimulation-based timed process algebras [9] proves effective forTACSlt, too.
Definition 7.7(weak MT-precongruence). A relationR ⊆ P × P is aweak MT-precongruence relation if, for allP, Q ∈ Randα∈ A:
(1) P −→α P implies∃Q, k, P, k. Q=σ⇒k α=⇒=σ⇒kQ,P−→σ k+kP, and P≈mtQ.
(2) Q−→α Q implies∃P. P =α⇒P andP≈mtQ. (3) P −→σ P implies∃Q. Q−→σ Q and P, Q ∈ R.
(4) Q−→σ Q implies∃P. P −→σ P andP, Q ∈ R.
We writePmtQifP, Q ∈ Rfor some weak MT-precongruence relationR, and callmt theweak MT-precongruence.
Again,mt is a preorder and the largest weak MT-precongruence relation. It is worth pointing out that the strong faster-than precongruence∼mtis contained in the weak faster-than precongruencemt, which follows by inspecting the respective definitions. The recursive definition of the weak MT-precongruence employed in
Conditions (3) and (4) above reflects the fact that clock transitions do not resolve choices [9].
Theorem 7.8. mt is the largest precongruence contained in≈mt.
Proof. The proof of compositionality of this preorder regarding theTACSlt oper-ators is quite standard, except for the parallel composition operator that needs to be treated as for the weak MT-preorder before. Containment is proved by showing thatmt∪≈mt is a weak MT-relation.
We are left with establishing the “largest” claim. From universal algebra we know that the largest precongruence ≈+mt in ≈mt exists and also that ≈+mt = {P, Q | ∀C[x]. C[P]≈mtC[Q]}. Since mt is a precongruence that is contained in ≈mt, the inclusion mt ⊆ ≈+mt holds. Thus, it remains to show ≈+mt ⊆ mt. Consider the relationauxmt =df{P, Q |P+c.0≈mtQ+c.0, wherec is not in the sorts ofP, Q}. By definition of auxmt we have≈+mt⊆ auxmt. We establish the other inclusionauxmt ⊆ mtby proving thatauxmt is a weak MT-precongruence relation.
LetPauxmt Q,i.e.,P +c.0≈mtQ+c.0, and distinguish the following cases.
• Action transitions: Let P −→α P, i.e., α = c and P +c.0 −→α P by Rule (Sum1). Since Pauxmt Q we know of the existence of some pro-cessesR, Pandk, k∈NsatisfyingQ+c.0=σ⇒k=α⇒ˆ =σ⇒kR,P −→σ k+kP andP≈mtR. SincePcannot perform ac-transition,Q+c.0must have performed some action fromQto becomeR; we concludeQ=σ⇒l α=⇒=σ⇒lR withl+l=k+k. The reverse case, where processQengages in an action transition, is straightforward, as Condition (2) of Definitions 7.2 and 7.7 coincides with the one for observation equivalence and observation con-gruence in CCS [20].
• Clock transitions: LetP −→σ P. By Rules (tAct) and (tSum),P+c.0−→σ P+c.0holds. SincePauxmtQwe know of the existence of some processR such that Q+c.0 −→σ R and P+c.0≈mtR. As clock derivatives are unique we haveR≡Q+c.0for someQ satisfyingQ−→σ Q. Becausec is a distinguished action not in the sorts of P and Q we may further concludePauxmt Q, as desired. The other case, where processQengages in a clock transition, is analogous.
This shows thatauxmt is a weak MT-precongruence relation. Hence,auxmt ⊆ mt,
as desired.
It remains an open question whether the weak MT-precongruence is also the largest precongruence contained in the weak amortized faster-than preorder.